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25 3 2/ nl n (n2)/ (nl)/ T Figure 5.2 Time diagram for reinvestment rates involving an investment of 1 at the end of each period for periods The accumulated value of this annuity is equal to the sum of the annuity payments and the accumulated value of the interest, i.e. n + i {Is)nrY\j = n + i (5.8) Formula (5.8) simplifies to the familiar if i = J. The consideration of reinvestment rates in financial calculations has become increasingly important and more widely used than heretofore. This reflects the greater volatility of interest rates in recent years, as well as the increased sophistication of investors. An important consideration to a lender (investor) is the rate of repayment by the borrower. The "faster" the rate of repayment, the more significant the reinvestment issue becomes. The "slower" the rate of repayment, the longer the initial investment rate will dominate the calculation. This phenomenon is illustrated in Example 5.6. An analytical approach to the measurement of the rate of repayment is developed in Section 9.8. One final observation is that the results of financial calculations involving reinvestment rates are dependent upon the period of time under consideration. Thus, it is important to specify the period of time for which calculations are being made when reinvestment rates are being taken into account. Example 5.5 Payments of $1000 are invested at the beginning of each ear for 10 years. The payments earn interest at 7% effective and the interest can be reinvested at 5% effective. (1) Find the amount in the fund at the end of 10 years. (2) Find the purchase price an investor should pay to produce a yield rate of 8% effective. 1. A variation of formula (5.8) appropriate for an annuitydue is TTtV  ( + 1) J 1000 10t.07  m.osii   == 1000  10+.07  14.206811        
= $14,490 to the nearest dollar. The answer lies between lOOOi. = $13,207 and lOOOijQi = $14,784 , as would be expected. 2. The purchase price to yield 8% effective would be 14,490(1.08)° = $6712 to the nearest dollar. Example 5.6 Compare the yield rates on the three loan repayment schedules described in Example 3.3, if the repayments to the lender can be reinvested at only 7% instead of 9% as earned on the original loan. 1. The accumulated value of all payments at the end of 10 years is 1000(1.09)° = $2367.36. The yield rate / is found from the equation of value 1000(1 + i)° = $2367.36 which is immediately seen to be / = .09. In this case, the reinvestment risk totally disappears, since the borrower makes no payments until die end of the period of the loan. 2. The accumulated value of all payments at the end of 10 years is 1000 + 90ijoo7 = 1000 + 90(13.8164) = $2243.48 by a direct appUcation of formula (5.7). The yield rate / is found from the equation of value 1000(1 +/)° = $2243.48 which gives / = .0842, or 8.42%. This answer is less dian die answer to case 1, since die reinvestment rate of 7% has an effect in diis case. iiol.07 = (155.82)(13.8164) = $2152.88 . 3. The accumulated value of all payments at the end of 10 years is 1000 «151.09 The yield rate / is found from the equation of value 1000(1 + /)° = $2152.88 which gives / = .0797, or 7.97%. This answer is less dian die answer to case 2, since the repayment schedule for case 3 is "faster" than case 2 which increases the
effect of the reinvestment rate on the answer. Note that the yield rate stiU exceeds 7%, as would be expected. 5.5 INTEREST MEASUREMENT OF A FUND A significant problem in practical work is the determination of th yield rate earned by an investment fund. Recall that the basic definition of an effective rate of interest given in Section 1.3 assumed that the principal remains constant throughout the period and that all the interest earned is paid at the end of the period. In practice these assumptions are often not satisfied. It is common for a fund to be incremented with new principal deposits, decremented with principal withdrawals, and incremented with interest earnings many times throughout a period, often at irregular intervals. Some method must be devised for these situations to determine reasonable effective rates of interest. Consider finding the effecfive rate of interest earned by a fund over one measurement period. We make the following definitions: (5.10) . = ab = the amount in the fund at the beginning of the period the amount in the fund at the end of the period the amount of interest earned during the period the net amount of principal contributed at fime t (positive or negative), where 0 < r < 1 the total net amount of principal contributed during the period (positive or negative), i.e. the amount of interest earned by 1 invested at time b over the following period of length a, where a > 0, b > 0, and a + b < \ Note that in terms of the notation used in Section 5.3 a = Bq and = B. We use the alternative notation involving a and in order to develop a formula in widespread use with its traditional notation. The fund at the end of the period must equal the fund at the beginning of the period plus net principal contributed (positive or negative) plus interest earned, i.e. = a + + i. (5.9) To be consistent with the definition of the effective rate of interest given in Section 1.3 we will assume that all the interest earned / is received at the end of the period. Then an exact equation of value for the interest earned over die period, 0 < r < 1, is Unfortunately, formula (5.10) is not in a form which can be directly solved for i. It is necessary to find values for Assuming compound interest throughout the period, we have = (1 +0"1. (5.11) We can substitute formula (5.11) into formula (5.10) obtaining an exact equation for /. This equation can be solved by iteration. Section 5.3 guarantees that the rate found by iteration will be unique as long as the fund balance never becomes negative. If a computer or pocket calculator with builtin financial functions is not available to do the iteration, or if only approximate answers are required, it is possible to produce a simplified formula by making the assumption that (5.12) Formula (5.12) is a version of simple interest for this situation. We can substihite formula (5.12) into formula (5.10) and solve for i, obtaining a + Yqnt) (5.13) The numerator of formula (5.13) is the amount of interest earned on the fund. The denominator can be interpreted as the average amount of principal invested and is often called the exposure associated with i. Although formula (5.13) does not produce a true effective rate of interest because of the simple interest assumption, it will generally produce results quite close to a true effective rate of interest as long as the Cs are small in relation to A, which is often the case in practice. However, if the C,s are not small in relation to A, then the error can become significant. ( Formula (5.13) is in a form which can be directiy calculated. However, the summation term in the denominator is often rather laborious. Therefore, a fiirther simplifying assumption is often made; namely, that principal deposits «nd withdrawals occur uniformly throughout the period. Thus, on average, we might assume that net principal contributions occur at time t = Ml. If this assumption is made, then formula (5.13) becomes A + .5C A + .5{BA i) from formula (5.9)
A+BI Figure (5.3) is an illustrative time diagram for this formula. A I (5.14) Figure 5.3 Time diagram for formula (5.14) Formula (5.14) is an important formula which is widely used in practice to calculate earned rates of interest, e.g. it has been used by insurance regulators to compute the yield rate on invested assets of insurance companies. It is a very convenient formula, since it involves only A, B, and /, which are readily available. However, it should be remembered that it does assume that net principal contributions occur at time r = 1/2. If this assumption is not warranted, then the more exact (but still approximate) formula (5.13) should be used. In some cases it is possible to develop simplified versions of formula (5.13) which will be more accurate than formula (5.14). For example, if it is Icnown that net principal contributions occur at time on the average, 0 < < 1, then a generalization of formula (5.14) is given by + (1  k)B  (1  k)[ (5.15) The derivation of formula (5.15) is left as an exercise. It is immediately clear that formula (5.15) becomes formula (5.14) when = 1/2. However, if we Icnow that net principal contributions occur on April 1 on the average, then for calendar year calculations the use of formula (5.15) with = 1/4 should produce superior answers to formula (5.14). Formula (5.12) looks very similar to simple interest as defined in Section 1.4. However, it can be shown that the two are not equivalent by considering the form of 5, under each assumption. As defined in Section 1.4, the accumulation function for simple interest is given by ait) = \+ti. (1.5) This is equivalent to the assumption that /0 = f (5.16) An expression for 6, under this assumption is given by formula (1.34) 5, = 1 +ti (1.34) For the version of simple interest defined by formula (5.12) we have = 1 + = 1 + (1 t)i 4dr = log,[l +(1 0] and differentiating with respect to t 5r = 1 +(1 t)i for 0 < / < 1 (5.17) Clearly formulas (1.34) and (5.17) are not equivalent; in fact, they are equal only for t  1/2. In addition, it should be noted that formula (1.34) is a decreasing function of /, while formula (5.17) is an increasing function of t. It is possible to develop analogous results for funds in which payments are being made continuously. Let B, be the outstanding fund balance at time t, 0 < f < n, and assume that contributions (positive or negative) are being made continuously at exact time t at the rate of C, per period. Then a generalized version of formula (5.6) is given by „=0(1+0"+ C,(l +/)"*. (5.18) In essence, formula (5.18) says that the fund balance at the end of n measurement periods is equal to the beginning fund balance accumulated with interest for n periods, plus the accumulated value of all the intervening payments (positive or negative) in the amount of C, dt accumulated with interest to the end of the n periods. An even more general formula can be obtained by also allowing the force of interest to vary continuously. A generalized version of formula (5.18) would then be given by S,ds 0 + S.ds (5.19) The following differential equation is associated with formula (5.19) 4B, = 5,B,+ C,. (5.20) Formula (5.20) has an interesting verbal interpretation. The lefthand side is the instantaneous rate of change in the fund balance at time t. The righthand side attributes this instantaneous rate of change to two factors: (1) interest at
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